一類2次多項式混沌系統(tǒng)的均勻化方法研究
doi: 10.11999/JEIT180735
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1.
北京科技大學數理學院 ??北京 ??100083
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2.
廈門大學嘉庚學院信息科學與技術學院 ??漳州 ??363105
基金項目: 中央高校基本科研業(yè)務費專項基金(06108236)
Homogenization Method for the Quadratic Polynomial Chaotic System
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1.
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
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2.
School of Information Science and Technology, Xiamen University Tan Kah Kee College, Zhangzhou 363105, China
Funds: The Fundamental Research Funds for the Central Universities of China (06108236)
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摘要: 該文給出了一般的2次多項式混沌系統(tǒng)與Tent映射拓撲共軛的充分條件，并依據該條件，給出了一類2次多項式混沌系統(tǒng)及其概率密度函數；進一步得到了能夠將這類系統(tǒng)均勻化的變換函數；給出了一個新的2次多項式混沌系統(tǒng)并進行均勻化處理，對其產生的序列進行了信息熵、Kolmogorov熵和離散熵分析，結果顯示該均勻化方法的均勻化效果顯著且不改變序列混沌程度。
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關鍵詞:
- 混沌系統(tǒng) /
- 均勻化 /
- 拓撲共軛 /
- 熵
Abstract: A sufficient condition for general quadratic polynomial systems to be topologically conjugate with Tent map is proposed. Base on this condition, the probability density function of a class of quadratic polynomial systems is provided and transformations function which can homogenize this class of chaotic systems is further obtained. The performances of both the original system and the homogenized system are evaluated. Numerical simulations show that the information entropy of the uniformly distributed sequences is closer to the theoretical limit and its discrete entropy remains unchanged. In conclusion, with such homogenization method all the chaotic characteristics of the original system is inherited and better uniformity is performed.-
Key words:
- Chaotic system /
- Homogenization /
- Topologically conjugate /
- Entropy
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表 1 幾個2次多項式混沌系統(tǒng)
混沌系統(tǒng)$f(x)$ 概率密度 均勻化系統(tǒng)$z(x)$ $f(x)$信息熵 $z(x)$信息熵 $f(x) = \frac{7}{2}{x^2} + \frac{{33}}{{10}}x - \frac{{53}}{{200}}$ $\frac{7}{{{\text{π}} \sqrt { - 49{x^2} + 46.2x + 5.11} }}$ $z(x) = \frac{1}{{\text{π}} }\arcsin \left( { - \frac{7}{4}x - \frac{{33}}{{40}}} \right)$ 8.6470 8.9651 $f(x) = \frac{5}{4}{x^2} - \frac{1}{2}x - \frac{{27}}{{20}}$ $\frac{5}{{{\text{π}} \sqrt { - 25{x^2} + 10x + 63} }}$ $z(x) = \frac{1}{{\text{π}} }\arcsin\left( { - \frac{5}{8}x + \frac{1}{8}} \right)$ 8.6380 8.9649 $f(x) = - \frac{5}{2}{x^2} + 3x + \frac{1}{2}$ $\frac{5}{{{\text{π}} \sqrt { - 25{x^2} + 30x + 7} }}$ $z(x) = \frac{1}{{\text{π}} }\arcsin\left( {\frac{5}{4}x - \frac{3}{4}} \right)$ 8.6412 8.9650 
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表 2 系統(tǒng)均勻化前后的信息熵與最大熵比較
$\left( {N,M} \right)$ 均勻化前
信息熵均勻化后
信息熵最大熵 (500000, 100) 6.3530 6.6437 6.6439 (500000, 300) 7.9137 8.2284 8.2288 (500000, 500) 8.6431 8.9651 8.9658
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